# The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

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Introduction

## Pattens in Squares

The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

In order to find the difference of n x n square first step is to take nxn square and then multiply the corners diagonally.

Middle

11 12

n+10 n+11 n2 +11n – n2 +11n +10

11 x 2 = 22 10

'font-size:14.0pt; '>12 x 1 = 12

'font-size:14.0pt; '>22 – 12 = 10

## In both ways the difference is 10

3x3 square

n n+2 n(n+22) – [(n+2) (n+20)]

1 2 3

11 12 13 n2+22n - n2+20n +2n +40

21 22 23

n+20 n+22 n2 +22n - n2 +22n +40

21 x 3 = 63 40

23 x 1 = 23

63 – 23 = 40

## In both ways the difference is 40

Conclusion

31 32 33 34 n2+33n - n2+33n + 90

n+30 n+33 10

## 31 x 4 = 124 In both ways the difference is 90

34 x 1 = 31

124 – 34 = 90

5x5 square

n n+4 n(n+44) - [(n+4) (n+40)]

1 2 3 4 5

11 12 13 14 15 n2+44n - n2+40n + 4n +160

21 22 23 24 25

31 32 33 34 35 n2+44n - n2+44n +160

41 42 43 44 45

n+40 n+44 160

41 x 5 = 205

45 x 1 = 45

205 – 45 = 160

## In both ways the difference is 160

6x6 square

n n+5 n(n+55) - [(n+5) (n+50)

1 2 3 4 5 6

11 12 13 14 15 16 n2+55n - n2+50n + 5n + 250

21 22 23 24 25 26

31 32 33 34 35 36 n2+55n - n2+55n + 250

41 42 43 44 45 46

51 52 53 54 55 56 250

n+50 n+55

51 x 6 = 306

56 x 1 = 56

306 – 56 = 250

## In both ways the difference is 250

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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